Euclid and the Sword

howardat58:

If you want to know more about Euclid and his influence, especially outside mathematics, read this. It is fascinating.

Originally posted on Boxing Pythagoras:

I have written, often, about one of my personal heroes from history, Euclid of Alexandria, who wrote a textbook called Elements which would serve as the foundation for all Western mathematics for 2000 years. You may recall that, outside of his name and a list of his writings, we know almost nothing about Euclid. We know nothing of his birth, or his schooling, or his politics. We don’t know if he traveled extensively or if he was relatively sedentary. We don’t know if he was tall, short, fat, skinny, handsome, or ugly. However, one thing we do know is that Euclid’s work, though purely mathematical, bore a tremendous influence on a wide variety of fields of knowledge.

Euclid’s Elements set out to prove the whole of mathematics deductively from very simple definitions, axioms, and postulates. Deductive logic provided a sound and absolute basis by which mathematics operated for every man…

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Radius of the inscribed circle of a right angled triangle

incircle radius with text

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Pythagoras, triples, 3,4,5, a calculator.

How to generate Pythagorean triples (example: 3,4,5), well one way at least.

Starting with (x + y)2 = x2 + y2 + 2xy and (x – y)2 = x2 + y2 – 2xy

we can write the difference of two squares

(x + y)2  –  (x – y)2 = 4xy

and if we write  x = A2 and y = B2 the right hand side is a square as well.

Thus:

(A2  +  B2) 2 – (A2 – B2) 2 = 4A2 B2 = (2AB) 2

which can be written as

(A2  +  B2) 2 = (A2 – B2) 2 + (2AB) 2

the Pythagoras form.

Now just put in some integers for A and B

2 and 1 gives 3,4,5

Conjecture1: This process generates ALL the Pythagorean triples.

Conjecture2: Every odd number belongs to some  Pythagorean triple.

Have fun…….

My next post will be about finding the radius of the inscribed circle in a right angled triangle…..

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“Observe and make use of structure”. Observe would be a start. A tale from the chalkface.

Here’s my little story:
It was a class of day release students on a Higher National Certificate course in engineering. I reached a point in one class with a relationship between p and q, p = kq, with k a constant. “What’s its graph look like”, I asked. Deathly silence. “Ok, let’s try x and y”. Result y = kx. Same question, same response. “Well, what about y = 3x ?”. Same question, same response. So I wrote y = 3x + 2. Their eyes lit up, and they unanimously shouted “It’s a straight line!”.

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CCSS SMP 7 Look for and make use of structure. Sums of powers of the natural numbers

The following is unreadable. Use the browser zoom or click the picture, or download the .doc file from https://howardat58.files.wordpress.com/2014/10/sums-of-powers.doc
sums of powers pic version

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The last performance – non mathematical in the formal sense.

In three hours from now (5.00pm PR time) takes place the last performance of Nana Badrena’s Ballet “DRACULA”, at the Teatro Yaguez in Mayaguez Puerto Rico. Created in 1999 in Cuba this ballet has been performed many times, in Cuba, Mexico, Pennsylvania, South America, and here. The ballet is based on the film from 1973 by Francis Ford Coppola.

This is our Ballet Company, Western Ballet Theatre, and we have a website http://www.wbtpr.com

(so it’s not just math !)(and theatre is the English spelling)
drac4Dracula seduces Lucy Westenra
drac6and then moves on to Mina, the reincarnation of his dead bride of 400 years earlier
drac7The vampires in Dracula’s castle.
drac3Jonathan, after Renfield grabs his valise
drac9The dramatic bed scene, as the drugged Jonathan wakes up.

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Halloween fun

Found on the net. With avocados raining down on my porch roof this seems a good use for them!

guacolantern

“To hell with pumpkins, try the Guaco-lantern

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Fractions, at home and away.

So one day recently I was bored, and then the following rushed onto the page:

Half of a big pizza is equal to two small pizzas – rewrite this in as precise way as possible.

one quarter

Is this 3 hours or 1/4 of a pizza?

How many hours equals 1/4 of a pizza?

Apologies to those for whom a clock face is a historical artifact.

How do I know it’s a pizza and not a cake?

Does half a day include half the night?

There are four feet in our yard, mine and my sister’s.

Would you prefer 1/2 of a round pizza or 1/2 of a square pizza?

Are ratios numbers or fractions (or neither) ?

Fractions are parts of the same whole. OK, I’ll have 5 quarters of that pizza (5/4 is a fraction, isn’t it?)

You cut, I choose !

This year fractions are parts of a whole. Next year fractions will be numbers. I guess the other party won the election.

2/3 and 4/6 are equivalent fractions. Equivalent to what ?

The word “fraction” has the same root as “fracture”. So something got broken. I think it was my faith in common sense.

The Common Core test question asked – How long is 3.25 hours. This could be 3 hours and 25 minutes or 3 hours and 15 minutes. I guess it depends on the grade level.

Back to the heavy stuff next time !

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Subtraction in algebra – let’s use algebra !

I have seen some heavy handed ways of explaining the identity

a – (b + c) = a – b – c

Let us use algebra. Give the left hand side a name, say d .Then

a – (b + c) = d

This is an equation, so add (b+c) to each side and get

a = d + (b + c), then a = d + b + c as the parentheses are now superfluous.

Now subtract  b  from each side

a – b = d + c

Now subtract  c  from each side

a – b – c = d

so  a – (b + c) =  a – b – c

or is this too simple ? Look, no messing with p – q = p + -(q) stuff,

and no appeal to the famous distributive law.

You can do this, and other stuff, with numbers as well.

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Calculus without limits 5: log and exp

The derivative of the log function can be investigated informally, as log(x) is seen as the inverse of the exponential function, written here as exp(x). The exponential function appears naturally from numbers raised to varying powers, but formal definitions of the exponential function are difficult to achieve. For example, what exactly is the meaning of exp(pi) or exp(root(2)).
So we look at the log function:-
calculus5 text
calculus5 pic

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